论文标题
可逆二阶张量的结构保存不变插值方案
Structure-Preserving Invariant Interpolation Schemes for Invertible Second-Order Tensors
论文作者
论文摘要
张量插值是在应用和科学学科的各个领域张量数据分析的重要步骤。在目前的工作中，提出了针对一般的新型插值方案，即对称或非对称，可逆的平方张量。至关重要的是，所提出的方案依赖于张量数据$ \ boldsymbol {t} \！\！= \！\！\ boldsymbol {r} \ boldsymbol {q}^t \！\！\！\！\！\！\！\！ \boldsymbolλ\ boldsymbol {q} $，然后是两个旋转张量$ \ boldsymbol {r} $和$ \ boldsymbol {q} $的单独插值，以及从这种decormosition decomportsions造成的decomptionts $ \ boldsymbol {q} $。考虑到特殊正交组$ \ mathbb {so}（3）$在基于相对旋转向量或四元组的特殊旋转插值中，考虑了两种不同的方案。对于特征值插值，考虑了三种不同的方案，即基于对数加权平均值，移动最小二乘或对数移动最小二乘。 It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, i.e., $\boldsymbol{R}$ and $\boldsymbol{Q}$ remain orthogonal tensors and $\boldsymbolΛ$ remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity).基于选择对称或非对称张量的插值的选定数值示例，将提出的方案与现有方法进行比较，例如欧几里得，log-euclidean，Cholesky和Log-cholesky插值。与这些现有方法相反，提出的插值方案导致张张量不变性的平稳而单调的演变，例如确定性，痕量，分数各向异性（FA）和Hilbert的各向异性（HA）... {继续见PDF}}
Tensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, i.e., symmetric or non-symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data $\boldsymbol{T}\!\!=\!\!\boldsymbol{R}\boldsymbol{Q}^T \!\! \boldsymbolΛ \boldsymbol{Q}$, followed by an individual interpolation of the two rotation tensors $\boldsymbol{R}$ and $\boldsymbol{Q}$ and the positive definite diagonal eigenvalue tensor $\boldsymbolΛ$ resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group $\mathbb{SO}(3)$, either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, i.e., $\boldsymbol{R}$ and $\boldsymbol{Q}$ remain orthogonal tensors and $\boldsymbolΛ$ remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non-symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log-Euclidean, Cholesky and Log-Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA)...{continued see pdf}
